Chapter 8
Scaling in Financial Time
Two well-documented findings motivate further analysis of financial
time series. First, the probability distributions of returns often deviate
significantly from the normal distribution by having fat tails and excess
kurtosis. Secondly, returns exhibit volatility clustering. The latter effect
has led to the development of the GARCH models described in Section
In this chapter, we shall focus on scaling in the probability distri-
butions of returns, the concept that has attracted significant attention
from economists and physicists alike.
Alas, as the leading experts in Econophysics, H. E. Stanley and
R. Mantegna acknowledged [2]:
‘‘No model exists for the stochastic process describing the
time evolution of the logarithm of price that is accepted by
all researchers.’’
There are several reasons for the status quo.
First, different financial
time series may have varying non-stationary components. Indeed, the
stock price reflects not only the current value of a company’s assets
but also the expectations of the company’s growth. Yet, there is no
general pattern for evolution of a business enterprise.

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